Let $G$ be a complex reductive group, $L\subset G$ a Levi subgroup and $Rep(G)$ the category of rational representations of $G$.

My Question: What is the geometric analogue of the restriction functor $Res^G_L:Rep(G)\to Rep(L)$?

To be a little bit more precise:

Let $\check{G}$ be the dual group of $G$. As usual define $\mathcal{K}:=\mathbb{C}$ and $\mathcal{O}:=\mathbb{C}[[t]]$. Let further $Gr_{\check{G}}:=\check{G}(\mathcal{K})/\check{G}(\mathcal{O})$ denote the affine Grassmannian and $P_{\check{G}(\mathcal{O})}(Gr_{\check{G}})$ the category of $\check{G}(\mathcal{O})$-equivariant perverse sheaves on $Gr$. The geometric Satake Isomorphism gives an equivalence of tensor categories categories $$P_{\check{G}(\mathcal{O})}(Gr_{\check{G}}) \cong Rep(G)$$

Is there a nice (from the geometric viewpoint) functor $\check{Res}:P_{\check{G}(\mathcal{O})}(Gr_{\check{G}})\to P_{\check{L}(\mathcal{O})}(Gr_{\check{L}})$ which corresponds under the geometric Satake isomorphism to the restriction functor $Res^G_L$?